As there is only $i=0$ for $j=n-1$ you can assume the $(n-1)$th series to be constant and continue all other series until you're at the series you were originally interested in.

About this method

I've heard that this is calculating the discrete derivate and it works for every function that is generated by a polynomial. But I'm not sure about that.

I am aware that this doesn't work like people expect (e.g. for the series that uses language like:

1
1 1
2 1
1 2 1 1

read like "one one" which gives the second row, "two ones", "one two, one one"...)

but it is a logical way to continue any sequence. When you want me to continue a sequence in a logical way but not with this method, you should provide more details about the sequence. However, when you come down to the last generated sequence with my method and have to assume that this one is constant, it's probably not the sequence that you should find.

Thanks it sounds interesting. Is this also usable for instance used to rule out that is "generated by a polynomial" ? btw what "polinomial generation" means exactly ? Would "polynomial" also include "simple" stuff like mod or ceiling/floor function ?
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PatternExplorerJul 7 '13 at 10:26

Thank you very much Zander. It certainly does give some interesting ideas to play with, and I will. (Let me know in case you have some more ideas, especially if not using recursion and even if not perfect match)
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PatternExplorerJul 8 '13 at 6:12